Difference between revisions of "Wave run-up"
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{{Definition|title=Wave run-up | {{Definition|title=Wave run-up | ||
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| − | [[File: | + | [[File:SetupRunup.jpg|thumb|400px|left|Fig. 1. Definition sketch wave set-down, wave set-up and wave run-up.]] |
| − | Wave run-up is the sum of [[wave set-up]] and | + | Wave run-up is the sum of [[wave set-up]] and [[swash]] uprush and must be added to the water level reached as a result of tides and wind set-up (Fig. 1). Wave run-up on a beach is generally due to so-called [[swash]] bores: the uprush of waves after final collapse on the beach. Wave run-up is an important parameter for assessing the safety of sea dikes or coastal settlements. |
| − | + | Waves refer to waves generated by wind (locally or on the ocean) or waves generated by incidental disturbances of the sea surface such as tsunamis, seiches or ship waves. Wave run-up is often indicated with the symbol <math> R </math>. | |
| − | For waves collapsing on the beach, a first order-of-magnitude estimate is given by the empirical formula of Hunt (1959) <ref>Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152</ref><ref>Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953</ref><ref>Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E. 2017. Assessment of runup predictions by empirical models on non-truncated beaches on the south-east Australian coast. Coast. Eng. 119: 15–31</ref>, | + | ==Empirical formulas== |
| + | For waves collapsing on the beach, a first order-of-magnitude estimate is given by the empirical formula of Hunt (1959) <ref name=H59>Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152</ref><ref>Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953</ref><ref name=A17>Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E. 2017. Assessment of runup predictions by empirical models on non-truncated beaches on the south-east Australian coast. Coast. Eng. 119: 15–31</ref>, | ||
| − | <math>R \sim \eta_u + H \xi , </math> | + | <math>R \sim \eta_u + H \xi , \qquad (1)</math> |
where <math>\eta_u \sim 0.2 H</math> is the [[wave set-up]], <math>H</math> is the offshore significant wave height and <math>\xi</math> is the [[surf similarity parameter]], | where <math>\eta_u \sim 0.2 H</math> is the [[wave set-up]], <math>H</math> is the offshore significant wave height and <math>\xi</math> is the [[surf similarity parameter]], | ||
| − | <math>\xi = \Large\frac{ | + | <math>\xi = \Large\frac{\beta}{\sqrt{H/L}}\normalsize = T \, \beta \, \Large\sqrt{\frac{g}{2\pi H}}\normalsize , \qquad (2)</math> |
| − | where <math>L = g T^2/(2 \pi)</math> | + | where <math>\tan \beta \approx \beta</math> is the beach slope and <math>T</math> is the peak wave period. The offshore wavelength <math>L</math> is given by <math>L = g T^2/(2 \pi)</math>, assuming that <math>L / (2 \pi) </math> is smaller than the local depth <math>h</math>. |
| − | The horizontal wave incursion is approximately given by <math> R / | + | The horizontal wave incursion is approximately given by <math> R / \beta</math>. |
| − | + | The maximum run-up depends on the mean water level at the shoreline. This water level is increased by wave-induced setup <math>\eta_0</math>, see [[Wave set-up]]. The following empirical relationship has been derived for the sum of maximum wave setup and swash runup<ref name=S6>Stockdon, H.F., Holman, R.A., Howd, P.A. and Sallenger, A.H. 2006. Empirical parameterization of setup, swash, and runup. Coast. Eng. 53: 573–588</ref>: | |
| − | <math> R_2 = 1.1 \; (\eta_u + 0.5 \sqrt{S_w^2 + S_{ig}^2} \, ) , \ | + | <math> R \approx 0.73 \; \beta \; \sqrt{HL} . \qquad (3)</math> |
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| + | Many empirical formulas have been proposed for the run-up. A popular formula for the run-up <math> R_2</math> exceeded by only 2 % of the waves for given <math>H, L</math> has been developed by Stockdon et al. (2006<ref name=S6>Stockdon, H.F., Holman, R.A., Howd, P.A. and Sallenger, A.H. 2006. Empirical parameterization of setup, swash, and runup. Coast. Eng. 53: 573–588</ref>), based on a large dataset: | ||
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| + | <math> R_2 = 1.1 \; (\eta_u + 0.5 \sqrt{S_w^2 + S_{ig}^2} \, ) , \qquad \xi \ge 0.3 , \qquad R_2= 0.043 \; \sqrt{HL} , \qquad \xi < 0.3 , \qquad (4) </math> | ||
where <math>\eta_u = 0.35 H \xi</math> is the wave set-up, <math>S_w=0.75 H \xi</math> is the swash uprush related to incident waves and <math>S_{ig}=0.06 \sqrt{HL}</math> is the additional uprush related to [[infragravity waves]]. The factor 1.1 takes into account the non-Gaussian distribution of run-up events. | where <math>\eta_u = 0.35 H \xi</math> is the wave set-up, <math>S_w=0.75 H \xi</math> is the swash uprush related to incident waves and <math>S_{ig}=0.06 \sqrt{HL}</math> is the additional uprush related to [[infragravity waves]]. The factor 1.1 takes into account the non-Gaussian distribution of run-up events. | ||
| − | + | An empirical expression taking the medium sediment grain size <math>d_{50}</math> [m] into account is<ref name=H59/><ref>Sunamura,T. 2004. A predictive relationship for the spacing of beach cusps in nature. Coastal Engineering 51: 697-711</ref>, | |
| − | The general applicability of empirical formulas | + | <math>R = 0.4 \beta \,T \, \sqrt{gH} \, \exp(-10 \, d_{50}^{0.55}) \, . \qquad (5) </math> |
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| + | This formula is less suitable for dissipative beaches, where swash is dominated by infragravity waves. | ||
| + | |||
| + | From an inventory of run-up formulas by Gomes da Silva et al. (2020<ref>Gomes da Silva, P., Coco, G., Garnier, R. and Klein, A.H.F. 2020. On the prediction of runup, setup and swash on beaches. Earth-Science Reviews 204, 103148</ref>), it appears that for steep beaches (<math> \beta > 0.1</math>) the run-up increases with increasing beach slope (approximately linear dependance<ref name=NH>Nielsen, P. and Hanslow, D.J. 1991. Wave runup distributions on natural beaches. J. Coast. Res. 7: 1139–1152</ref>), while for gently sloping dissipative beaches (<math> \beta < 0.1</math>) the dependence on beach slope is weak or absent<ref name=NH/><ref name=S6/>. In these latter cases, run-up is dominated by [[infragravity waves]], that yield a small run-up that increases with increasing wave height (approximately linear dependence<ref>Ruessink, B.G., Kleinhans, M.G. and Van Den Beukel, P.G.L. 1998. Observations of swash under highly dissipative conditions. J. Geophys. Res. 103: 3111–3118</ref><ref>Ruggiero, P., Holman, R. A. and Beach, R. A. 2004. Wave run-up on a high-energy dissipative beach. J. Geophys. Res. 109, C06025, doi:10.1029/2003JC002160</ref>). Field observations<ref>Matsuba, Y. and Shimozono, T. 2021. Analysis of the contributing factors to infragravity swash based on long-term observations. Coastal Engineering 169, 103957</ref> and numerical models<ref>Guza, R.T. and Feddersen, F. 2012. Effect of wave frequency and directional spread on shoreline runup. Geophys. Res. Lett. 39: 1–5</ref> point to a dependence of infragravity [[swash]] on the frequency spread and the directional spread of incident waves. The largest infragravity swash has been observed for incident waves with a small directional spread and a large frequency spread. | ||
| + | |||
| + | ==Run-up on cobble revetments== | ||
| + | Blenkinsopp et al. (2022<ref>Blenkinsopp, C.E., Bayle, P.M., Martins, K., Foss, O.W., Almeida, L.-P., Kaminsky, G.M., Schimmels, S. and Matsumoto, H. 2022. Wave runup on composite beaches and dynamic cobble berm revetments. Coastal Engineering 176, 104148</ref>) observed that the significant swash height increases substantially as the swash zone moves from the lower dissipative sand beach to a higher reflective gravel berm during rising tide. In cases where the backshore berm or foredune of such composite beaches is (artificially) protected with gravel or cobbles, the run-up <math> R_2</math> exceeded by only 2 % of the waves when the water level during storms reaches higher than the toe of the cobble berm, can be estimated from the approximate empirical formula | ||
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| + | <math> R_2 \approx 0.26 + 0.19 H + 3.1 H_{toe} \, \beta_{berm} \, . \qquad (6) </math> | ||
| + | |||
| + | The first two terms represent the wave setup; <math>\beta_{berm}</math> is the berm slope and <math>H_{toe}</math> is the significant wave height at the toe of the berm. For saturated (depth-limited) surf zones, the significant wave height at the toe of the berm can be estimated from the [[breaker index]] <math>\gamma_b</math> and the water depth <math>h_{toe}</math> at the toe, <math>H_{toe} = \gamma_b h_{toe}</math>. See also the article [[Wave overtopping]]. | ||
| + | |||
| + | ==Accuracy of run-up estimates== | ||
| + | The accuracy (root-mean-square deviation) of run-up predictions using empirical formulas such as (1), (3) or (4) is usually not better than <math>0.25 \, R</math>.<ref name=A17/> The general applicability of empirical formulas for run-up prediction based on simple parametric representations of beach and shoreface is limited due to the influence of the more detailed characteristics of the local beach and shoreface bathymetry (e.g., beach permeability and groundwater level, curvature of the beach profile, crescentic bars, or swash acceleration along the horn of a [[Beach cusps|beach cusp]] embayment<ref>Kim, L.N., Brodie, K.L., Cohn, N.T., Giddings, S.N. and Merrifield, M. 2023. Observations of beach change and runup, and the performance of empirical runup parameterizations during large storm events. Coastal Engineering 184, 104357</ref>). The tide level may influence the wave run-up due to wave breaking on [[nearshore sandbars]].<ref>Guedes, R.M.C., Bryan, K.R., Coco, G. and Holman, R.A. 2011. The effects of tides on swash statistics on an intermediate beach. J. Geophys. Res. 116, C04008</ref> The applicability of empirical formulas for the [[wave set-up]], which is a substantial component of the run-up, is limited for similar reasons. Accurate estimates of the wave run-up require in-situ observations or detailed numerical models. | ||
| + | |||
| + | A more detailed discussion of wave run-up and backwash is given in the articles [[Swash]] and [[Swash zone dynamics]]. | ||
| + | |||
| + | An analytical model estimate of the runup of monochromatic non-broken waves on a uniform sloping beach is discussed in the article [[Waves on a sloping bed]]. | ||
| + | |||
| + | ==Run-up saturation== | ||
| + | The empirical formulas suggest that the run-up is an ever increasing function of the incident wave height. Field and laboratory observations show that this is not the case<ref name=G84>Guza, R.T., Thornton, E.B. and Holman, R.A. 1984. Swash on steep and shallow beaches. Proceedings of the 19th International Conference Coastal Engineering, ASCE, 1, pp. 708 – 723</ref><ref name=H17>Hughes, M.G., Baldock, T.E. and Aagaard, T. 2017. Swash saturation: is it universal and do we have an appropriate model? Coastal Dynamics Conf. 2017, Paper No.108, pp. 192-203</ref>. Several processes limit the uprush on the beach when incident waves are very high: breaking of short waves in the surf zone (leading to surf zone saturation), breaking of infragravity waves on the beach face and collision of the uprushing swash bore with the backwash of the preceding swash bore. More details can be found in the articles [[Swash]], [[Swash zone dynamics]] and [[Breaker index]]. The greatest wave run-up occurs for the highest waves that do not break, which are generally waves in the lower part of the wave frequency spectrum (long-period waves). Theoretical formulas for run-up saturation (for monochromatic frictionless waves, see the articles [[Waves on a sloping bed]], [[Swash]]) have the form | ||
| + | |||
| + | <math>R < R_s = A \, g \, \beta^2 \, T^2 \, . \qquad (7)</math> | ||
| + | |||
| + | Using for the coefficient <math>A</math> the value <math>A \approx 0.14</math>, reasonable agreement is found with field and laboratory observations. <ref name=H17/> | ||
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: [[Wave set-up]] | : [[Wave set-up]] | ||
: [[Swash]] | : [[Swash]] | ||
| + | : [[Waves on a sloping bed]] | ||
: [[Tsunami]] | : [[Tsunami]] | ||
| + | : [[Breaker index]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
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| + | |||
| + | {{author | ||
| + | |AuthorID=120 | ||
| + | |AuthorFullName=Job Dronkers | ||
| + | |AuthorName=Dronkers J}} | ||
| + | |||
| + | [[Category:Beaches]] | ||
Latest revision as of 21:17, 11 January 2025
Definition of Wave run-up:
Wave run-up is the maximum onshore elevation reached by waves, relative to the shoreline position in the absence of waves.
This is the common definition for Wave run-up, other definitions can be discussed in the article
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Wave run-up is the sum of wave set-up and swash uprush and must be added to the water level reached as a result of tides and wind set-up (Fig. 1). Wave run-up on a beach is generally due to so-called swash bores: the uprush of waves after final collapse on the beach. Wave run-up is an important parameter for assessing the safety of sea dikes or coastal settlements.
Waves refer to waves generated by wind (locally or on the ocean) or waves generated by incidental disturbances of the sea surface such as tsunamis, seiches or ship waves. Wave run-up is often indicated with the symbol [math] R [/math].
Contents
Empirical formulas
For waves collapsing on the beach, a first order-of-magnitude estimate is given by the empirical formula of Hunt (1959) [1][2][3],
[math]R \sim \eta_u + H \xi , \qquad (1)[/math]
where [math]\eta_u \sim 0.2 H[/math] is the wave set-up, [math]H[/math] is the offshore significant wave height and [math]\xi[/math] is the surf similarity parameter,
[math]\xi = \Large\frac{\beta}{\sqrt{H/L}}\normalsize = T \, \beta \, \Large\sqrt{\frac{g}{2\pi H}}\normalsize , \qquad (2)[/math]
where [math]\tan \beta \approx \beta[/math] is the beach slope and [math]T[/math] is the peak wave period. The offshore wavelength [math]L[/math] is given by [math]L = g T^2/(2 \pi)[/math], assuming that [math]L / (2 \pi) [/math] is smaller than the local depth [math]h[/math]. The horizontal wave incursion is approximately given by [math] R / \beta[/math].
The maximum run-up depends on the mean water level at the shoreline. This water level is increased by wave-induced setup [math]\eta_0[/math], see Wave set-up. The following empirical relationship has been derived for the sum of maximum wave setup and swash runup[4]:
[math] R \approx 0.73 \; \beta \; \sqrt{HL} . \qquad (3)[/math]
Many empirical formulas have been proposed for the run-up. A popular formula for the run-up [math] R_2[/math] exceeded by only 2 % of the waves for given [math]H, L[/math] has been developed by Stockdon et al. (2006[4]), based on a large dataset:
[math] R_2 = 1.1 \; (\eta_u + 0.5 \sqrt{S_w^2 + S_{ig}^2} \, ) , \qquad \xi \ge 0.3 , \qquad R_2= 0.043 \; \sqrt{HL} , \qquad \xi \lt 0.3 , \qquad (4) [/math]
where [math]\eta_u = 0.35 H \xi[/math] is the wave set-up, [math]S_w=0.75 H \xi[/math] is the swash uprush related to incident waves and [math]S_{ig}=0.06 \sqrt{HL}[/math] is the additional uprush related to infragravity waves. The factor 1.1 takes into account the non-Gaussian distribution of run-up events.
An empirical expression taking the medium sediment grain size [math]d_{50}[/math] [m] into account is[1][5],
[math]R = 0.4 \beta \,T \, \sqrt{gH} \, \exp(-10 \, d_{50}^{0.55}) \, . \qquad (5) [/math]
This formula is less suitable for dissipative beaches, where swash is dominated by infragravity waves.
From an inventory of run-up formulas by Gomes da Silva et al. (2020[6]), it appears that for steep beaches ([math] \beta \gt 0.1[/math]) the run-up increases with increasing beach slope (approximately linear dependance[7]), while for gently sloping dissipative beaches ([math] \beta \lt 0.1[/math]) the dependence on beach slope is weak or absent[7][4]. In these latter cases, run-up is dominated by infragravity waves, that yield a small run-up that increases with increasing wave height (approximately linear dependence[8][9]). Field observations[10] and numerical models[11] point to a dependence of infragravity swash on the frequency spread and the directional spread of incident waves. The largest infragravity swash has been observed for incident waves with a small directional spread and a large frequency spread.
Run-up on cobble revetments
Blenkinsopp et al. (2022[12]) observed that the significant swash height increases substantially as the swash zone moves from the lower dissipative sand beach to a higher reflective gravel berm during rising tide. In cases where the backshore berm or foredune of such composite beaches is (artificially) protected with gravel or cobbles, the run-up [math] R_2[/math] exceeded by only 2 % of the waves when the water level during storms reaches higher than the toe of the cobble berm, can be estimated from the approximate empirical formula
[math] R_2 \approx 0.26 + 0.19 H + 3.1 H_{toe} \, \beta_{berm} \, . \qquad (6) [/math]
The first two terms represent the wave setup; [math]\beta_{berm}[/math] is the berm slope and [math]H_{toe}[/math] is the significant wave height at the toe of the berm. For saturated (depth-limited) surf zones, the significant wave height at the toe of the berm can be estimated from the breaker index [math]\gamma_b[/math] and the water depth [math]h_{toe}[/math] at the toe, [math]H_{toe} = \gamma_b h_{toe}[/math]. See also the article Wave overtopping.
Accuracy of run-up estimates
The accuracy (root-mean-square deviation) of run-up predictions using empirical formulas such as (1), (3) or (4) is usually not better than [math]0.25 \, R[/math].[3] The general applicability of empirical formulas for run-up prediction based on simple parametric representations of beach and shoreface is limited due to the influence of the more detailed characteristics of the local beach and shoreface bathymetry (e.g., beach permeability and groundwater level, curvature of the beach profile, crescentic bars, or swash acceleration along the horn of a beach cusp embayment[13]). The tide level may influence the wave run-up due to wave breaking on nearshore sandbars.[14] The applicability of empirical formulas for the wave set-up, which is a substantial component of the run-up, is limited for similar reasons. Accurate estimates of the wave run-up require in-situ observations or detailed numerical models.
A more detailed discussion of wave run-up and backwash is given in the articles Swash and Swash zone dynamics.
An analytical model estimate of the runup of monochromatic non-broken waves on a uniform sloping beach is discussed in the article Waves on a sloping bed.
Run-up saturation
The empirical formulas suggest that the run-up is an ever increasing function of the incident wave height. Field and laboratory observations show that this is not the case[15][16]. Several processes limit the uprush on the beach when incident waves are very high: breaking of short waves in the surf zone (leading to surf zone saturation), breaking of infragravity waves on the beach face and collision of the uprushing swash bore with the backwash of the preceding swash bore. More details can be found in the articles Swash, Swash zone dynamics and Breaker index. The greatest wave run-up occurs for the highest waves that do not break, which are generally waves in the lower part of the wave frequency spectrum (long-period waves). Theoretical formulas for run-up saturation (for monochromatic frictionless waves, see the articles Waves on a sloping bed, Swash) have the form
[math]R \lt R_s = A \, g \, \beta^2 \, T^2 \, . \qquad (7)[/math]
Using for the coefficient [math]A[/math] the value [math]A \approx 0.14[/math], reasonable agreement is found with field and laboratory observations. [16]
Related articles
References
- ↑ 1.0 1.1 Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152
- ↑ Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953
- ↑ 3.0 3.1 Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E. 2017. Assessment of runup predictions by empirical models on non-truncated beaches on the south-east Australian coast. Coast. Eng. 119: 15–31
- ↑ 4.0 4.1 4.2 Stockdon, H.F., Holman, R.A., Howd, P.A. and Sallenger, A.H. 2006. Empirical parameterization of setup, swash, and runup. Coast. Eng. 53: 573–588
- ↑ Sunamura,T. 2004. A predictive relationship for the spacing of beach cusps in nature. Coastal Engineering 51: 697-711
- ↑ Gomes da Silva, P., Coco, G., Garnier, R. and Klein, A.H.F. 2020. On the prediction of runup, setup and swash on beaches. Earth-Science Reviews 204, 103148
- ↑ 7.0 7.1 Nielsen, P. and Hanslow, D.J. 1991. Wave runup distributions on natural beaches. J. Coast. Res. 7: 1139–1152
- ↑ Ruessink, B.G., Kleinhans, M.G. and Van Den Beukel, P.G.L. 1998. Observations of swash under highly dissipative conditions. J. Geophys. Res. 103: 3111–3118
- ↑ Ruggiero, P., Holman, R. A. and Beach, R. A. 2004. Wave run-up on a high-energy dissipative beach. J. Geophys. Res. 109, C06025, doi:10.1029/2003JC002160
- ↑ Matsuba, Y. and Shimozono, T. 2021. Analysis of the contributing factors to infragravity swash based on long-term observations. Coastal Engineering 169, 103957
- ↑ Guza, R.T. and Feddersen, F. 2012. Effect of wave frequency and directional spread on shoreline runup. Geophys. Res. Lett. 39: 1–5
- ↑ Blenkinsopp, C.E., Bayle, P.M., Martins, K., Foss, O.W., Almeida, L.-P., Kaminsky, G.M., Schimmels, S. and Matsumoto, H. 2022. Wave runup on composite beaches and dynamic cobble berm revetments. Coastal Engineering 176, 104148
- ↑ Kim, L.N., Brodie, K.L., Cohn, N.T., Giddings, S.N. and Merrifield, M. 2023. Observations of beach change and runup, and the performance of empirical runup parameterizations during large storm events. Coastal Engineering 184, 104357
- ↑ Guedes, R.M.C., Bryan, K.R., Coco, G. and Holman, R.A. 2011. The effects of tides on swash statistics on an intermediate beach. J. Geophys. Res. 116, C04008
- ↑ Guza, R.T., Thornton, E.B. and Holman, R.A. 1984. Swash on steep and shallow beaches. Proceedings of the 19th International Conference Coastal Engineering, ASCE, 1, pp. 708 – 723
- ↑ 16.0 16.1 Hughes, M.G., Baldock, T.E. and Aagaard, T. 2017. Swash saturation: is it universal and do we have an appropriate model? Coastal Dynamics Conf. 2017, Paper No.108, pp. 192-203
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